- Email: [email protected]
Transitivity of fuzzy preference relations – an empirical study
Fuzzy Sets and Systems 118 (2001) 503–508
www.elsevier.com/locate/fss
Transitivity of fuzzy preference relations – an empirical study ∗ Ã Zbigniew Switalski Department of Operations Research, University of Economics, al.NiepodlegloÃsci 10, 60-967 PoznaÃn, Poland Received March 1997; received in revised form May 1998
Abstract Based on an experiment we show that human preferences, when represented by fuzzy relations, may violate transitivity. c 2001 Elsevier Science We deÿne an index of transitivity and compute the degree to which the transitivity is violated. B.V. All rights reserved. Keywords: Empirical research; Decision making; Fuzzy preferences
1. Introduction There is a lot of evidence showing that human preferences, when represented by crisp or probabilistic relation may violate transitivity (see e.g. [3,5] and [1,4,7] for more recent references). When we use a crisp model we can confront a subject with two alternatives: A and B and ask him which of them is better for him (in a given situation). In this case we allow only three possibilities for answer: “A is better than B”, “B is better than A”, “A is equivalent to B”. In a probabilistic model we ask a subject many times “which is better: A or B?”. Then the proportion of answers “A better than B” with respect to all answers may be treated as a degree (probability) in which A is better than B: P(A; B) number of answers “A better than B” : = number of all answers Financially supported by Polish Committee on Scientiÿc Research – grant No. P 110 039 06. ∗ Tel.: + 48-61-822-6356.
We can use also fuzzy model of preferences and using it we ask a subject “To what extent A is better than B”. As an answer we should obtain a number from the unit interval [0; 1], say R(A; B), representing subjective judgement of degree in which A is better than B (intensity of preference). When we use a crisp model we can meet intransitivities in a set of answers i.e. for a given subject there can be three alternatives, say A; B and C, such that A is better than B; B is better than C and C is better than A for this person. When probabilistic model is used there can be observed violations from the following weak condition of transitivity (weak stochastic transitivity): (P(A; B)¿ 12 ∧ P(B; C)¿ 12 ) ⇒ P(A; C)¿ 12 and of course from stronger conditions (moderate stochastic transitivity, strong stochastic transitivity, see [3]). We can formulate the following problem: what (if any) kind of intransitivities will we observe when a fuzzy model is used and if we observe an intransitivity what will be the degree of this intransitivity (in
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 8 7 - 5
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
504
the fuzzy model, as well as in the probabilistic one, we can measure in the appropriate way the degree of intransitivity). In Section 4 we report results of an experiment in which 44 students had to deÿne their fuzzy preferences among ÿve objects. We study occuring intransitivities from the point of view of di erent deÿnitions of transitivity for fuzzy relations and measure in all cases deviation from transitivity (we do not want to explain in any way observed intransitivities, our aim is only to state the results of an experiment, compare e.g. [1,4,7] for di erent explanations of intransitivity of preferences in crisp case).
Fig. 1.
We will consider also two additional transitivity conditions: (4) -transitivity ( ∈ [0; 1]): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿ max(R(A; B); R(B; C)) + (1 − ) min(R(A; B); R(B; C)):
2. Transitivity properties for fuzzy (valued) relations
(5) G-transitivity: R(A; C)¿R(A; B) + R(B; C) − 1:
We will use (as in the probabilistic model) fuzzy relations i.e. functions R : X 2 → [0; 1]; where X is a ÿxed (ÿnite) set of objects (alternatives), satisfying R(A; B) + R(B; A) = 1 for all A; B ∈ X; A 6= B. Hence, if we ask a subject “what is the degree of preference of A over B” and obtain an answer R(A; B), then we automatically obtain R(B; A) as 1 − R(A; B). So we can say that A is better than B if R(A; B) ¿ 12 (and R(A; B) is the measure of intensity of preference), B is better than A if R(A; B) ¡ 12 (and 1 − R(A; B) = R(B; A) measures intensity of this preference) and R(A; B) = 12 means equivalence of alternatives A and B. We deÿne the following transitivity conditions (all formulas are valid for every A; B; C ∈ X ), see [3,5]: (1) Strong stochastic transitivity (S-transitivity): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿max(R(A; B); R(B; C)):
(S)
(2) Moderate stochastic transitivity (M-transitivity): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿min(R(A; B); R(B; C)):
(G)
Condition ( ) is an intermediate condition between strong and moderate transitivity. Its idea is taken from [2]. The G-transitivity may be called group transitivity because if n subjects have preferences Ri which are crisp linear orders i.e. for all A; B; C ∈ X : (i) Ri (A; B) ∈ {0; 1}; (ii) Ri (A; A) = 0; (iii) Ri (A; B) + Ri (B; A) = 1; A 6= B; (iv) Ri (A; C)¿Ri (A; B) + Ri (B; C) − 1; then the convex combination of Ri deÿned by ! X X i Ri (A; B) i ¿0; i = 1 ; R(A; B) = i
i
(1) which may be interpreted as a relation of group preference ( i is weight associated with ith subject) satisÿes condition (G). It can be checked that the G-transitivity is equivalent to the condition min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿R(A; B) + R(B; C) − 1
(M)
(3) Weak stochastic transitivity (W-transitivity): min(R(A; B); R(B; C))¿ 12 ⇒ R(A; C)¿ 12 :
( )
(W)
which is similar to (1) – (4). Fig. 1 shows all logical relationships between deÿned transitivity conditions. It is worth noting that there is no relationship between properties (W) and (G).
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
505
3. An experiment In our experiment, 44 students of the Faculty of Management, University of Economics in PoznaÃn ÿlled in the following questionnaire (the idea of “fuzzy questionnaire” was taken from [6]): “Imagine that you can spend your holidays in one (and only one) of the following countries: Finland (Fi), France (Fr), Germany (G), Italy (I), Scotland (S). Fix your preferences with respect to every pair of countries in the following manner: for every pair of countries, say A and B, you have an interval of length 10 cm, with ends A and B:
Fix a point in this interval (by putting a sign “X ”) representing your preferences such that if you ÿx the point A, then you deÿnitely prefer spending your holidays in A than in B, if you ÿx B, then you deÿnitely prefer B than A, if you ÿx the middle point of [A; B], then you are being indi erent to where you want to spend your holidays – in A or in B, and if you put the sign “X ” on the right part of the interval, then you prefer B than A and the nearer is “X ” to B the stronger is preference of B over A (analogously the left part of [A; B] denotes preference of A over B)”. Students obtained ten intervals (one after another) chosen in random order. For every answer the distance (in mm) from “X ” to B was measured (denoted by d(A; B)) and thus we obtained for every student the preference matrix consisting of numbers R(A; B) = 0:01d(A; B) and R(B; A) = 1 − R(A; B). All obtained data are given in the appendix. To study transitivity of obtained preference relations we considered triples {A; B; C} of elements from the set {Fi; Fr; G; I; S}. We say that a triple {A; B; C} is S-transitive (M-, W-, -, G-transitive) for a relation R if R is S- (M-, W-, -, G-) transitive in the set {A; B; C} i.e. if the condition (S) ((M), (W), ( ), (G)) is satisÿed for every ordering of elements A; B; C. Examples of transitive triples are presented in Fig. 2. The triple in Fig. 2a is S-transitive, in Fig. 2b is M-transitive and -transitive for 6(0:75 − 0:71)= (0:82 − 0:71) = 0:36, in Fig. 2c is W-transitive and G-transitive, in Fig. 2d is W-transitive and not G-transitive, in Fig. 2e is G-transitive and not W-
Fig. 2.
transitive, in Fig. 2f is not G-transitive nor Wtransitive. The di erence between cases (e) and (f) is that in the case (e) we have R(A; B) + R(B; C) + R(C; A)62 (this condition satisÿed for every ordered triple (A; B; C) is equivalent to G-transitivity) and in the case (f) this inequality is not satisÿed. To measure the degree in which transitivity is satisÿed we will deÿne the following index of transitivity. Let (A; B; C) be an ordered triple such that min(R(A; B); R(B; C))¿ 12 and let TR be the set of all such triples for a relation R in a set X . For (A; B; C) ∈ TR we let t(A; B; C) = 1 − max(max(R(A; B); R(B; C)) −R(A; C); 0): For any 3-element set {A1 ; A2 ; A3 } (i.e. an unordered triple) we let t{A1 ; A2 ; A3 } = min t(Ai ; Aj ; Ak ); (i; j; k)∈K
where K is the set of all permutations (i; j; k) of {1; 2; 3} such that (Ai ; Aj ; Ak ) ∈ TR . For example, in the case (a) in Fig. 1 t{A; B; C} = 1 (t = 1 for all S-transitive triples), in the case (b) t{A; B; C}=
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
506
Table 1 Number of triples and number of subjects satisfying given transitivity condition
Table 2 Number of triples and number of subjects satisfying inequality t¿ÿ
Kind of transitivity
Number of triples
%
Number of subjects
%
Inequality
Number of triples
%
Number of subjects
%
S-transitivity 0.9-transitivity 0.8-transitivity 0.7-transitivity 0.6-transitivity 0.5-transitivity 0.4-transitivity 0.3-transitivity 0.2-transitivity 0.1-transitivity M-transitivity W-transitivity G-transitivity
309 332 347 363 372 381 391 394 398 402 412 430 436
70.2 75.5 78.9 82.5 84.6 86.6 88.9 89.6 90.5 91.4 93.6 97.7 99.1
4 4 6 12 14 17 20 20 21 22 25 35 42
9.1 9.1 13.6 27.3 31.8 38.6 45.5 45.5 47.8 50 56.8 79.6 95.5
t¿0:95 t¿0:9 t¿0:85 t¿0:8 t¿0:7 t¿0:6 t¿0:5
358 389 411 420 427 437 438
81.4 88.4 93.4 95.5 97.1 99.3 99.6
9 19 26 31 35 42 43
20.5 43.2 59.1 70.5 79.6 95.5 97.7
0:93, in the case (c) t{A; B; C} = 0:83, in the case (d) t{A; B; C} = 0:69 (in cases (a) – (d) the set K consists of one element – (A; B; C)), in the case (e) t{A; B; C} = 0:51 (in this case K = {(A; B; C); (B; C; A); (C; A; B)}; t(A; B; C) = 0:71; t(B; C; A) = 0:51; t(C; A; B) = 0:51), in the case (f) t{A; B; C} = 0:47. 4. Results In the set {Fi; Fr; G; I; S} we have 10 triples, so for 44 subjects we obtained 440 triples. For every triple the kind of transitivity satisÿed by this triple was determined and the index of transitivity was computed. In Table 1 the number of triples and the number of subjects satisfying given transitivity condition is presented (a subject satisÿes the given transitivity condition if all ten triples for his preference relation satisfy this condition). In Table 2 the number of triples and subjects satisfying inequality t¿ÿ, is given. 5. Conclusions It appears that, in general, preferences in our experiment were not transitive (= not S-transitive). In other words, almost every subject had at least one non-transitive triple. On the other hand, the number
of intransitive triples for particular subjects was rather small (30% on average). Of course, when weakening transitivity conditions, the number of triples and subjects satisfying the givencondition is increasing as it is shown in Table 1. For example the “cyclical” intransitivity: R(A; B) ¿ 12 ∧ R(B; C) ¿ 12 ∧ R(A; C) ¿ 12 was very rarely observed. The degree in which transitivity is satisÿed can be measured by some “fuzzy” index of transitivity (t). In our case, as it is shown in Table 2, the index t is in most cases near 1 and so the “distance” from transitivity is rather small. We can say that, although the preferences are in many cases intransitive, yet the “distance” from transitivity is, in general, rather small. We can also observe (see the appendix) that only two subjects (Nos. 28 and 33) presented their preferences as crisp relations (i.e. R(A; B) = 0 or 1 for all A; B), although in our experiment there was absolute freedom to choose any number between 0 and 1. We may conclude that people (when they have possibility to choose) prefer to describe their preferences as fuzzy ones not crisp and that vagueness is quite a common phenomenon when we ask about preferences. We can suppose that the results of similar experiments can depend on the kind of questions posed, set of alternatives considered, set of subjects, kind of transitivities and from the particular type of preference relation chosen (for example we could remove the condition R(A; B) + R(B; A) = 1). Further research could give some insight on these dependencies.
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
We could also think about reasons causing observed phenomena and try to build models (probabilistic, multicriterial or others) explaining the occuring of intransitivities in (fuzzy) preferences.
507
S13 = (14; 9; 17; 21; 66; 38; 74; 38; 35; 78); S14 = (19; 50; 8; 28; 50; 9; 23; 0; 21; 91); S15 = (8; 26; 3; 29; 38; 11; 85; 8; 76; 100); S16 = (0; 16; 9; 0; 87; 85; 32; 17; 11; 14); S17 = (0; 18; 0; 0; 100; 81; 90; 18; 0; 9);
Acknowledgements
S18 = (19; 51; 52; 30; 83; 72; 33; 47; 26; 41);
I would like to thank Krzysztof Piasecki and two anonymous referees for their comments that helped me to improve my paper.
S19 = (12; 36; 87; 15; 97; 100; 72; 95; 14; 0); S20 = (0; 39; 7; 42; 100; 100; 100; 11; 21; 51); S21 = (0; 27; 0; 0; 73; 22; 21; 25; 20; 52); S22 = (36; 74; 100; 28; 100; 100; 54; 100; 0; 0);
Appendix
S23 = (29; 61; 7; 6; 87; 18; 8; 3; 0; 52); S24 = (0; 33; 70; 0; 95; 100; 5; 72; 8; 0);
Here we present preference relations (for every subject) obtained in our experiment. Preferences for subject Si form a vector
S26 = (100; 100; 0; 100; 77; 0; 32; 0; 25; 100);
Si = (ai1 ; ai2 ; ai3 ; ai4 ; ai5 ; ai6 ; ai7 ; ai8 ; ai9 ; ai10 );
S27 = (0; 39; 0; 33; 82; 48; 62; 0; 22; 35);
where ai1 = Ri (Fi; Fr);
ai6 = Ri (Fr; S);
ai2 = Ri (Fi; G);
ai7 = Ri (Fr; I );
ai3
= Ri (Fi; S);
ai8
= Ri (G; S);
= Ri (Fi; I );
ai9
= Ri (G; I );
ai4
ai5 = Ri (Fr; G);
ai10 = Ri (S; I ):
For notation convenience all numbers are multiplied by 100. The data are as follows: S1 = (11; 84; 64; 0; 100; 84; 60; 7; 0; 19); S2 = (0; 8; 12; 0; 82; 88; 0; 49; 0; 0); S3 = (10; 79; 11; 10; 100; 73; 92; 9; 7; 43); S4 = (0; 16; 100; 0; 100; 100; 0; 100; 0; 0); S5 = (57; 65; 51; 64; 100; 67; 100; 0; 23; 52); S6 = (27; 50; 47; 74; 66; 59; 88; 50; 51; 70); S7 = (14; 24; 0; 0; 86; 30; 50; 0; 14; 49); S8 = (18; 79; 24; 16; 91; 51; 50; 16; 13; 9);
S25 = (0; 35; 54; 0; 100; 80; 36; 24; 0; 0);
S28 = (0; 100; 100; 0; 100; 100; 0; 0; 0; 0); S29 = (30; 31; 28; 32; 68; 83; 67; 63; 21; 30); S30 = (0; 30; 61; 0; 79; 100; 0; 45; 0; 0); S31 = (41; 49; 50; 6; 71; 71; 5; 36; 6; 7); S32 = (31; 100; 15; 22; 100; 77; 49; 0; 0; 16); S33 = (0; 100; 0; 0; 100; 100; 0; 0; 0; 0); S34 = (15; 100; 0; 11; 100; 0; 15; 0; 0; 48); S35 = (49; 100; 74; 55; 100; 67; 72; 10; 0; 46); S36 = (33; 7; 57; 59; 100; 55; 78; 0; 20; 59); S37 = (18; 100; 34; 13; 65; 30; 0; 0; 0; 22); S38 = (10; 38; 60; 13; 88; 92; 56; 30; 22; 12); S39 = (100; 100; 80; 91; 60; 29; 38; 20; 23; 48); S40 = (7; 41; 42; 0; 93; 70; 47; 36; 0; 0); S41 = (7; 49; 17; 9; 94; 88; 58; 37; 8; 5); S42 = (14; 74; 37; 34; 100; 65; 51; 8; 10; 26); S43 = (9; 50; 26; 12; 92; 75; 52; 28; 19; 12); S44 = (26; 75; 63; 24; 86; 81; 57; 14; 20; 21):
S9 = (35; 51; 52; 20; 77; 82; 19; 80; 23; 4); S10 = (42; 14; 0; 0; 49; 14; 0; 9; 1; 45); S11 = (76; 97; 43; 96; 86; 4; 95; 7; 49; 97); S12 = (6; 0; 27; 51; 18; 84; 84; 100; 100; 82);
References [1] M. Bar-Hillel, A. Margalit, How vicious are cycles of intransitive choice, Theory and Decision 24 (1988) 119–145.
508
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
[2] K. Basu, Fuzzy revealed preference theory, J. Econom. Theory 32 (1984) 212–227. [3] C.H. Coombs, R.M. Dawes, A. Tversky, Mathematical Psychology. An Elementary Introduction, Prentice-Hall, Englewood Cli s, NJ, 1970. [4] W.V. Gehrlein, The expected likelihood of transitivity – a survey, Theory and Decision 37 (1994) 175–209. [5] R.D. Luce, P. Suppes, Preference, utility and subjective probability, in: R.D. Luce, R.R. Busch, E. Galanter (Eds.),
Handbook of Math. Psychology, vol. III, Wiley, New York, 1965. [6] Cz. Noworol, Cluster Analysis in Empirical Research – Fuzzy Hierarchical Models, PWN, Warszawa, 1989 (in Polish). [7] B. Sopher, G. Gigliotti, Intransitive cycles: rational choice or random error, Theory and Decision 35 (1993) 311–336.
www.elsevier.com/locate/fss
Transitivity of fuzzy preference relations – an empirical study ∗ Ã Zbigniew Switalski Department of Operations Research, University of Economics, al.NiepodlegloÃsci 10, 60-967 PoznaÃn, Poland Received March 1997; received in revised form May 1998
Abstract Based on an experiment we show that human preferences, when represented by fuzzy relations, may violate transitivity. c 2001 Elsevier Science We deÿne an index of transitivity and compute the degree to which the transitivity is violated. B.V. All rights reserved. Keywords: Empirical research; Decision making; Fuzzy preferences
1. Introduction There is a lot of evidence showing that human preferences, when represented by crisp or probabilistic relation may violate transitivity (see e.g. [3,5] and [1,4,7] for more recent references). When we use a crisp model we can confront a subject with two alternatives: A and B and ask him which of them is better for him (in a given situation). In this case we allow only three possibilities for answer: “A is better than B”, “B is better than A”, “A is equivalent to B”. In a probabilistic model we ask a subject many times “which is better: A or B?”. Then the proportion of answers “A better than B” with respect to all answers may be treated as a degree (probability) in which A is better than B: P(A; B) number of answers “A better than B” : = number of all answers Financially supported by Polish Committee on Scientiÿc Research – grant No. P 110 039 06. ∗ Tel.: + 48-61-822-6356.
We can use also fuzzy model of preferences and using it we ask a subject “To what extent A is better than B”. As an answer we should obtain a number from the unit interval [0; 1], say R(A; B), representing subjective judgement of degree in which A is better than B (intensity of preference). When we use a crisp model we can meet intransitivities in a set of answers i.e. for a given subject there can be three alternatives, say A; B and C, such that A is better than B; B is better than C and C is better than A for this person. When probabilistic model is used there can be observed violations from the following weak condition of transitivity (weak stochastic transitivity): (P(A; B)¿ 12 ∧ P(B; C)¿ 12 ) ⇒ P(A; C)¿ 12 and of course from stronger conditions (moderate stochastic transitivity, strong stochastic transitivity, see [3]). We can formulate the following problem: what (if any) kind of intransitivities will we observe when a fuzzy model is used and if we observe an intransitivity what will be the degree of this intransitivity (in
c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 9 8 ) 0 0 2 8 7 - 5
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
504
the fuzzy model, as well as in the probabilistic one, we can measure in the appropriate way the degree of intransitivity). In Section 4 we report results of an experiment in which 44 students had to deÿne their fuzzy preferences among ÿve objects. We study occuring intransitivities from the point of view of di erent deÿnitions of transitivity for fuzzy relations and measure in all cases deviation from transitivity (we do not want to explain in any way observed intransitivities, our aim is only to state the results of an experiment, compare e.g. [1,4,7] for di erent explanations of intransitivity of preferences in crisp case).
Fig. 1.
We will consider also two additional transitivity conditions: (4) -transitivity ( ∈ [0; 1]): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿ max(R(A; B); R(B; C)) + (1 − ) min(R(A; B); R(B; C)):
2. Transitivity properties for fuzzy (valued) relations
(5) G-transitivity: R(A; C)¿R(A; B) + R(B; C) − 1:
We will use (as in the probabilistic model) fuzzy relations i.e. functions R : X 2 → [0; 1]; where X is a ÿxed (ÿnite) set of objects (alternatives), satisfying R(A; B) + R(B; A) = 1 for all A; B ∈ X; A 6= B. Hence, if we ask a subject “what is the degree of preference of A over B” and obtain an answer R(A; B), then we automatically obtain R(B; A) as 1 − R(A; B). So we can say that A is better than B if R(A; B) ¿ 12 (and R(A; B) is the measure of intensity of preference), B is better than A if R(A; B) ¡ 12 (and 1 − R(A; B) = R(B; A) measures intensity of this preference) and R(A; B) = 12 means equivalence of alternatives A and B. We deÿne the following transitivity conditions (all formulas are valid for every A; B; C ∈ X ), see [3,5]: (1) Strong stochastic transitivity (S-transitivity): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿max(R(A; B); R(B; C)):
(S)
(2) Moderate stochastic transitivity (M-transitivity): min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿min(R(A; B); R(B; C)):
(G)
Condition ( ) is an intermediate condition between strong and moderate transitivity. Its idea is taken from [2]. The G-transitivity may be called group transitivity because if n subjects have preferences Ri which are crisp linear orders i.e. for all A; B; C ∈ X : (i) Ri (A; B) ∈ {0; 1}; (ii) Ri (A; A) = 0; (iii) Ri (A; B) + Ri (B; A) = 1; A 6= B; (iv) Ri (A; C)¿Ri (A; B) + Ri (B; C) − 1; then the convex combination of Ri deÿned by ! X X i Ri (A; B) i ¿0; i = 1 ; R(A; B) = i
i
(1) which may be interpreted as a relation of group preference ( i is weight associated with ith subject) satisÿes condition (G). It can be checked that the G-transitivity is equivalent to the condition min(R(A; B); R(B; C))¿ 12
⇒ R(A; C)¿R(A; B) + R(B; C) − 1
(M)
(3) Weak stochastic transitivity (W-transitivity): min(R(A; B); R(B; C))¿ 12 ⇒ R(A; C)¿ 12 :
( )
(W)
which is similar to (1) – (4). Fig. 1 shows all logical relationships between deÿned transitivity conditions. It is worth noting that there is no relationship between properties (W) and (G).
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
505
3. An experiment In our experiment, 44 students of the Faculty of Management, University of Economics in PoznaÃn ÿlled in the following questionnaire (the idea of “fuzzy questionnaire” was taken from [6]): “Imagine that you can spend your holidays in one (and only one) of the following countries: Finland (Fi), France (Fr), Germany (G), Italy (I), Scotland (S). Fix your preferences with respect to every pair of countries in the following manner: for every pair of countries, say A and B, you have an interval of length 10 cm, with ends A and B:
Fix a point in this interval (by putting a sign “X ”) representing your preferences such that if you ÿx the point A, then you deÿnitely prefer spending your holidays in A than in B, if you ÿx B, then you deÿnitely prefer B than A, if you ÿx the middle point of [A; B], then you are being indi erent to where you want to spend your holidays – in A or in B, and if you put the sign “X ” on the right part of the interval, then you prefer B than A and the nearer is “X ” to B the stronger is preference of B over A (analogously the left part of [A; B] denotes preference of A over B)”. Students obtained ten intervals (one after another) chosen in random order. For every answer the distance (in mm) from “X ” to B was measured (denoted by d(A; B)) and thus we obtained for every student the preference matrix consisting of numbers R(A; B) = 0:01d(A; B) and R(B; A) = 1 − R(A; B). All obtained data are given in the appendix. To study transitivity of obtained preference relations we considered triples {A; B; C} of elements from the set {Fi; Fr; G; I; S}. We say that a triple {A; B; C} is S-transitive (M-, W-, -, G-transitive) for a relation R if R is S- (M-, W-, -, G-) transitive in the set {A; B; C} i.e. if the condition (S) ((M), (W), ( ), (G)) is satisÿed for every ordering of elements A; B; C. Examples of transitive triples are presented in Fig. 2. The triple in Fig. 2a is S-transitive, in Fig. 2b is M-transitive and -transitive for 6(0:75 − 0:71)= (0:82 − 0:71) = 0:36, in Fig. 2c is W-transitive and G-transitive, in Fig. 2d is W-transitive and not G-transitive, in Fig. 2e is G-transitive and not W-
Fig. 2.
transitive, in Fig. 2f is not G-transitive nor Wtransitive. The di erence between cases (e) and (f) is that in the case (e) we have R(A; B) + R(B; C) + R(C; A)62 (this condition satisÿed for every ordered triple (A; B; C) is equivalent to G-transitivity) and in the case (f) this inequality is not satisÿed. To measure the degree in which transitivity is satisÿed we will deÿne the following index of transitivity. Let (A; B; C) be an ordered triple such that min(R(A; B); R(B; C))¿ 12 and let TR be the set of all such triples for a relation R in a set X . For (A; B; C) ∈ TR we let t(A; B; C) = 1 − max(max(R(A; B); R(B; C)) −R(A; C); 0): For any 3-element set {A1 ; A2 ; A3 } (i.e. an unordered triple) we let t{A1 ; A2 ; A3 } = min t(Ai ; Aj ; Ak ); (i; j; k)∈K
where K is the set of all permutations (i; j; k) of {1; 2; 3} such that (Ai ; Aj ; Ak ) ∈ TR . For example, in the case (a) in Fig. 1 t{A; B; C} = 1 (t = 1 for all S-transitive triples), in the case (b) t{A; B; C}=
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
506
Table 1 Number of triples and number of subjects satisfying given transitivity condition
Table 2 Number of triples and number of subjects satisfying inequality t¿ÿ
Kind of transitivity
Number of triples
%
Number of subjects
%
Inequality
Number of triples
%
Number of subjects
%
S-transitivity 0.9-transitivity 0.8-transitivity 0.7-transitivity 0.6-transitivity 0.5-transitivity 0.4-transitivity 0.3-transitivity 0.2-transitivity 0.1-transitivity M-transitivity W-transitivity G-transitivity
309 332 347 363 372 381 391 394 398 402 412 430 436
70.2 75.5 78.9 82.5 84.6 86.6 88.9 89.6 90.5 91.4 93.6 97.7 99.1
4 4 6 12 14 17 20 20 21 22 25 35 42
9.1 9.1 13.6 27.3 31.8 38.6 45.5 45.5 47.8 50 56.8 79.6 95.5
t¿0:95 t¿0:9 t¿0:85 t¿0:8 t¿0:7 t¿0:6 t¿0:5
358 389 411 420 427 437 438
81.4 88.4 93.4 95.5 97.1 99.3 99.6
9 19 26 31 35 42 43
20.5 43.2 59.1 70.5 79.6 95.5 97.7
0:93, in the case (c) t{A; B; C} = 0:83, in the case (d) t{A; B; C} = 0:69 (in cases (a) – (d) the set K consists of one element – (A; B; C)), in the case (e) t{A; B; C} = 0:51 (in this case K = {(A; B; C); (B; C; A); (C; A; B)}; t(A; B; C) = 0:71; t(B; C; A) = 0:51; t(C; A; B) = 0:51), in the case (f) t{A; B; C} = 0:47. 4. Results In the set {Fi; Fr; G; I; S} we have 10 triples, so for 44 subjects we obtained 440 triples. For every triple the kind of transitivity satisÿed by this triple was determined and the index of transitivity was computed. In Table 1 the number of triples and the number of subjects satisfying given transitivity condition is presented (a subject satisÿes the given transitivity condition if all ten triples for his preference relation satisfy this condition). In Table 2 the number of triples and subjects satisfying inequality t¿ÿ, is given. 5. Conclusions It appears that, in general, preferences in our experiment were not transitive (= not S-transitive). In other words, almost every subject had at least one non-transitive triple. On the other hand, the number
of intransitive triples for particular subjects was rather small (30% on average). Of course, when weakening transitivity conditions, the number of triples and subjects satisfying the givencondition is increasing as it is shown in Table 1. For example the “cyclical” intransitivity: R(A; B) ¿ 12 ∧ R(B; C) ¿ 12 ∧ R(A; C) ¿ 12 was very rarely observed. The degree in which transitivity is satisÿed can be measured by some “fuzzy” index of transitivity (t). In our case, as it is shown in Table 2, the index t is in most cases near 1 and so the “distance” from transitivity is rather small. We can say that, although the preferences are in many cases intransitive, yet the “distance” from transitivity is, in general, rather small. We can also observe (see the appendix) that only two subjects (Nos. 28 and 33) presented their preferences as crisp relations (i.e. R(A; B) = 0 or 1 for all A; B), although in our experiment there was absolute freedom to choose any number between 0 and 1. We may conclude that people (when they have possibility to choose) prefer to describe their preferences as fuzzy ones not crisp and that vagueness is quite a common phenomenon when we ask about preferences. We can suppose that the results of similar experiments can depend on the kind of questions posed, set of alternatives considered, set of subjects, kind of transitivities and from the particular type of preference relation chosen (for example we could remove the condition R(A; B) + R(B; A) = 1). Further research could give some insight on these dependencies.
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
We could also think about reasons causing observed phenomena and try to build models (probabilistic, multicriterial or others) explaining the occuring of intransitivities in (fuzzy) preferences.
507
S13 = (14; 9; 17; 21; 66; 38; 74; 38; 35; 78); S14 = (19; 50; 8; 28; 50; 9; 23; 0; 21; 91); S15 = (8; 26; 3; 29; 38; 11; 85; 8; 76; 100); S16 = (0; 16; 9; 0; 87; 85; 32; 17; 11; 14); S17 = (0; 18; 0; 0; 100; 81; 90; 18; 0; 9);
Acknowledgements
S18 = (19; 51; 52; 30; 83; 72; 33; 47; 26; 41);
I would like to thank Krzysztof Piasecki and two anonymous referees for their comments that helped me to improve my paper.
S19 = (12; 36; 87; 15; 97; 100; 72; 95; 14; 0); S20 = (0; 39; 7; 42; 100; 100; 100; 11; 21; 51); S21 = (0; 27; 0; 0; 73; 22; 21; 25; 20; 52); S22 = (36; 74; 100; 28; 100; 100; 54; 100; 0; 0);
Appendix
S23 = (29; 61; 7; 6; 87; 18; 8; 3; 0; 52); S24 = (0; 33; 70; 0; 95; 100; 5; 72; 8; 0);
Here we present preference relations (for every subject) obtained in our experiment. Preferences for subject Si form a vector
S26 = (100; 100; 0; 100; 77; 0; 32; 0; 25; 100);
Si = (ai1 ; ai2 ; ai3 ; ai4 ; ai5 ; ai6 ; ai7 ; ai8 ; ai9 ; ai10 );
S27 = (0; 39; 0; 33; 82; 48; 62; 0; 22; 35);
where ai1 = Ri (Fi; Fr);
ai6 = Ri (Fr; S);
ai2 = Ri (Fi; G);
ai7 = Ri (Fr; I );
ai3
= Ri (Fi; S);
ai8
= Ri (G; S);
= Ri (Fi; I );
ai9
= Ri (G; I );
ai4
ai5 = Ri (Fr; G);
ai10 = Ri (S; I ):
For notation convenience all numbers are multiplied by 100. The data are as follows: S1 = (11; 84; 64; 0; 100; 84; 60; 7; 0; 19); S2 = (0; 8; 12; 0; 82; 88; 0; 49; 0; 0); S3 = (10; 79; 11; 10; 100; 73; 92; 9; 7; 43); S4 = (0; 16; 100; 0; 100; 100; 0; 100; 0; 0); S5 = (57; 65; 51; 64; 100; 67; 100; 0; 23; 52); S6 = (27; 50; 47; 74; 66; 59; 88; 50; 51; 70); S7 = (14; 24; 0; 0; 86; 30; 50; 0; 14; 49); S8 = (18; 79; 24; 16; 91; 51; 50; 16; 13; 9);
S25 = (0; 35; 54; 0; 100; 80; 36; 24; 0; 0);
S28 = (0; 100; 100; 0; 100; 100; 0; 0; 0; 0); S29 = (30; 31; 28; 32; 68; 83; 67; 63; 21; 30); S30 = (0; 30; 61; 0; 79; 100; 0; 45; 0; 0); S31 = (41; 49; 50; 6; 71; 71; 5; 36; 6; 7); S32 = (31; 100; 15; 22; 100; 77; 49; 0; 0; 16); S33 = (0; 100; 0; 0; 100; 100; 0; 0; 0; 0); S34 = (15; 100; 0; 11; 100; 0; 15; 0; 0; 48); S35 = (49; 100; 74; 55; 100; 67; 72; 10; 0; 46); S36 = (33; 7; 57; 59; 100; 55; 78; 0; 20; 59); S37 = (18; 100; 34; 13; 65; 30; 0; 0; 0; 22); S38 = (10; 38; 60; 13; 88; 92; 56; 30; 22; 12); S39 = (100; 100; 80; 91; 60; 29; 38; 20; 23; 48); S40 = (7; 41; 42; 0; 93; 70; 47; 36; 0; 0); S41 = (7; 49; 17; 9; 94; 88; 58; 37; 8; 5); S42 = (14; 74; 37; 34; 100; 65; 51; 8; 10; 26); S43 = (9; 50; 26; 12; 92; 75; 52; 28; 19; 12); S44 = (26; 75; 63; 24; 86; 81; 57; 14; 20; 21):
S9 = (35; 51; 52; 20; 77; 82; 19; 80; 23; 4); S10 = (42; 14; 0; 0; 49; 14; 0; 9; 1; 45); S11 = (76; 97; 43; 96; 86; 4; 95; 7; 49; 97); S12 = (6; 0; 27; 51; 18; 84; 84; 100; 100; 82);
References [1] M. Bar-Hillel, A. Margalit, How vicious are cycles of intransitive choice, Theory and Decision 24 (1988) 119–145.
508
à Z. Switalski / Fuzzy Sets and Systems 118 (2001) 503–508
[2] K. Basu, Fuzzy revealed preference theory, J. Econom. Theory 32 (1984) 212–227. [3] C.H. Coombs, R.M. Dawes, A. Tversky, Mathematical Psychology. An Elementary Introduction, Prentice-Hall, Englewood Cli s, NJ, 1970. [4] W.V. Gehrlein, The expected likelihood of transitivity – a survey, Theory and Decision 37 (1994) 175–209. [5] R.D. Luce, P. Suppes, Preference, utility and subjective probability, in: R.D. Luce, R.R. Busch, E. Galanter (Eds.),
Handbook of Math. Psychology, vol. III, Wiley, New York, 1965. [6] Cz. Noworol, Cluster Analysis in Empirical Research – Fuzzy Hierarchical Models, PWN, Warszawa, 1989 (in Polish). [7] B. Sopher, G. Gigliotti, Intransitive cycles: rational choice or random error, Theory and Decision 35 (1993) 311–336.